On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain

∂ttu − ∆u + λ |x|2 u ≥ |u| p in (0,∞) × Ω, where Ω is the exterior of the unit ball in RN , N ≥ 2, p > 1, and λ ≥ − (N−2 2 )2, under the inhomogeneous boundary condition α ∂u ∂ν (t, x) + βu(t, x) ≥ w(x) on (0,∞) × ∂Ω, where α, β ≥ 0 and (α, β) ≠ (0, 0). Namely, we show that there exists a critical exponent pc(N, λ) ∈ (1,∞] for which, if 1 < p < pc(N, λ), the above problem admits no global weak solution for any w ∈ L1(∂Ω) with ∫ ∂Ω w(x) dσ > 0, while if p > pc(N, λ), the problem admits global solutions for some w > 0. To the best of our knowledge, the study of the critical behavior for wave inequalities with a Hardy potential in an exterior domain was not considered in previous works. Some open questions are also mentioned in this paper.


Introduction
In this paper, we are concerned with the study of existence and nonexistence of global weak solutions to the wave inequality In the special case V ≡ , (1.3) has been investigated by several authors. Namely, John [12] proved that, if the initial values are compactly supported and nonnegative, then for N = and < p < pc( ) = + √ , nontrivial solutions must blow-up in nite time, while if p > pc( ), global solutions exist for small initial values. Next, a similar result has been derived by Glassey [6] in the case N = . In [19], Sha er proved that in the case N ∈ { , }, pc(N) belongs to the blow-up case. Georgiev et al. [5] (see also [15,21]) proved that, if p > pc (N) and N ≥ , then global solutions exist for small initial values. A blow-up result was shown by Sideris [20] (see also [9,18]) in the case < p < pc(N) and N ≥ . In [23], Yordanov and Zhang proved that for all N ≥ , pc(N) belongs to the blow-up case.
In [22], Yordanov and Zhang studied (1.3) when N ≥ and V is a nonnegative potential satisfying the following conditions: "There exist functions ϕ i ∈ C (R N ), i = , , such that e |x| with positive constants C i , i = , ". It was shown that, if the initial values are nonnegative and compactly supported, then a blow-up occurs when < p < pc(N).
In [7], Hamidi and Laptev considered semilinear evolution inequalities of the form The study of blow-up phenomena for semilinear wave equations in exterior domains was considered by many authors (see e.g. [8,10,11,13,14,24,25] and the references therein). In particular, Zhang [24] studied the semilinear wave equation under the inhomogeneous Neumann boundary condition 6) where N ≥ , w ∈ L (∂Ω), w ≥ , and w ≢ . Namely, it was shown that (1.5)-(1.6) admits as critical exponent the real number p * = + N− , i.e. if < p < p * , then (1.5)-(1.6) admits no global weak solution, while if p > p * , global solutions exist for some w > . Later, the same critical exponent was obtained for (1.5) under the inhomogeneous Dirichlet boundary condition [10] and the Robin boundary condition [8] ∂u To enlarge the literature review on the main topic of this article, we recall the study of blow-up of solutions carried out by Mohammed et al. [17], for fully nonlinear uniformly elliptic equations. Also, we mention the recent work of Bahrouni et al. [1], where the authors dealt with a class of double phase variational functionals related to the study of transonic ow, and established useful integral inequalities. In a series of remarkable papers, Cîrstea and Rădulescu [2][3][4] focused on special classes of semilinear elliptic equations (namely, logistic equations) and linked the nonregular variation of the nonlinearity at in nity with the blow-up rate of the solutions. They also established existence and uniqueness results for related problems, in the cases of homogeneous Dirichlet, Neumann or Robin boundary condition.
To the best of our knowledge, the study of critical behavior for wave inequalities with Hardy potential in an exterior domain was not considered in previous works. In this paper, we investigate the critical behavior for (1.1) under the inhomogeneous boundary condition (1.2). Namely, we will show that there exists a critical exponent pc(N, λ) ∈ ( , ∞] for which, when < p < pc(N, λ) and ∂Ω w(x) dσ > , (1. We introduce the test function space where C c (O) denotes the space of C functions compactly supported in O. Notice that Ω is closed and ∂O ⊂ O.
Now, we are ready to state our main results. We discuss separately the cases λ = − N− and λ > − N− .
From Theorems 1.1 and 1.2, one deduces that, The above statements show that the exponent pc(N, λ) is critical for (1.1)-(1.2).

Remark 1.2. From Theorems 1.1 and 1.2, we deduce that pc(N, λ) is also critical for the exterior problem
(1.10) The rest of the paper is organized as follows. In Section 2, we establish some lemmas and provide some estimates that will be used in the proofs of our main results. Section 3 is devoted to the proofs of Theorems 1.1 and 1.2. Namely, we rst prove the nonexistence results (parts (i) and (ii) of Theorem 1.1, and part (i) of Theorem 1.2), next we prove the existence results (part (iii) of Theorem 1.1 and part (ii) of Theorem 1.2).

Preliminaries
For λ ≥ − N− , let ∆ λ be the di erential operator de ned by For α, β ≥ and (α, β) ≠ ( , ), we introduce the function H α,β de ned in Ω by One can check easily that H α,β is a nonnegative solution to the exterior problem We need also to introduce two cut-o functions. Let η, ξ ∈ C ∞ (R) be such that where ≥ and θ > are constants to be chosen later. Proof. It can be easily seen that φ T ≥ , and for su ciently large T, φ T ∈ C c (O). On the other hand, for < |x| < + ϵ (ϵ > is su ciently small), one has By the de nition of the cut-o function ξ , since T is supposed to be large enough, one obtains Similarly, one has Next, we take α = . If λ = − N− , for r = |x|, one has Hence, if α = , in both cases, we have ∂H T ∂ν ∂Ω < , which yields (since η ≥ ) ∂φ T ∂ν ∂O ≤ , and the lemma is proved.
Throughout this paper, C denotes a positive constant (independent of T) whose value may change from line to line.

Lemma 2.2. For all < T < ∞ and ≥ , we have
Proof. By the de nition of the function η T , and using the properties of the cut-o function η, one obtains and the lemma is proved.

Lemma 2.3. Let m > . For all < T < ∞ and ≥ m m− , we have
Proof. It can be easily seen that |η ′′ which yields the desired estimate. So, if β = and N = , one obtains Proof. In this case, one has Hence, for su ciently large T, we obtain which yields the desired estimate.
Proof. For all x ∈ Ω, one has where " · " denotes the inner product in R N . Since ∆ λ H ( ) α,β = , it holds that Hence, it holds that where Now, let us estimate I i (T), i = , . Using the properties of the cut-o function ξ , for su ciently large T, one has On the other hand, it can be easily seen that for < |y| < √ , one has Hence, it holds that Hence, in both cases, by (2.6), for su ciently large T, one deduces that Next, one has It can be easily seen that for < |y| < √ , one has |∇ ξ (|y| ) | ≤ Cξ (|y| ) − .
Hence, it holds that

. Nonexistence results
We prove below parts (i) and (ii) of Theorem 1.1, as well as part (i) of Theorem 1.2. The proof is based on a rescaled test-function argument (see [16] for a general account of these methods) and a judicious choice of the test function.
Proof. Let us suppose that u ∈ L p loc (O) is a global weak solution to (1.1)-(1.2). By (1.9), we obtain for every φ ∈ Φ α,β . Using ε-Young inequality with ε = , we get Hence, it follows from (3.1), (3.2) and (3.3) that for every φ ∈ Φ α,β . By Lemma 2.1 and (3.4), for all ≥ p p− , θ > , and su ciently large T, one has Now, we shall estimate the terms from the right-hand side of the above inequality. By the de nition of the function φ T , one has On the other hand, using Lemma 2.3 with m = p, we obtain Moreover, combining Lemma 2.4 with Lemma 2.5, one deduces that for all λ ≥ − N− , Hence, by (3.6), (3.7) and (3.8), it holds that Again, by the de nition of the function φ T , one has Consider now the term from the left-hand side of (3.5). By the de nition of Lφ T , if α > , one has By the de nition of the function H T , and using Lemma 2.2, it holds that Since T is supposed to be large enough, by the de nition of the cut-o function ξ , we get On the other hand, by the de nition of the function H α,β , for all x ∈ ∂Ω (|x| = ), one has Then, for all λ ≥ − N− , one obtains If α = , by the de nition of Lφ T , and using Lemma 2.2, one has Notice that by (2.1) and (2.2), one has Hence, for all λ ≥ − N− , one obtains (3.14) Combining (3.13) with (3.14), one obtains Since < δ < , taking < ε < δ( −δ)